Mathematics
This domain tests PK–3 mathematical content and the pedagogy of number, place value, operations, fractions, algebra, geometry, measurement, and data. Items often describe a student error and ask you to diagnose the reasoning and choose an instructional response.
Number and counting
- Counting rests on stable order, one-to-one correspondence, and cardinality. Subitizing small sets and conservation of number underpin later work.
- Place value is a bundling system: ten ones make one ten, ten tens make one hundred. Students compose and decompose numbers with manipulatives and drawings before they meet the standard algorithm.
Operations and fact fluency
- Build meaning through joining, separating, and comparing situations, then strategies: make ten, doubles and near doubles, derived facts, and decomposing a number to bridge ten.
- Word-problem types include result, change, and start unknown, plus comparison problems. Key-word rules fail; draw strip diagrams or equations instead.
- Multiplication and division begin as equal-group and array models.
Fractions, geometry, and measurement
- Fractions develop from part-whole, set, and number-line models. Unit fractions anchor comparison: one-third is larger than one-fourth because the whole is split into fewer pieces.
- Geometry includes naming, composing, and decomposing two- and three-dimensional shapes; measurement moves from direct comparison to units, then measurement with rulers; data work begins with tally, picture, and bar graphs.
Algebraic thinking
- Patterns, open sentences, relational thinking, and the idea that the equals sign means "is the same as," not "compute."
Worked example
A student says one-fifth is greater than one-third "because 5 is bigger than 3." That is whole-number bias. The remedy is not a rule to memorize but experiences comparing unit fractions with fraction strips and number lines, naming the denominator as the number of equal parts, and benchmarking against one-half.
Traps
- Introducing algorithms before place-value understanding.
- Teaching key words instead of situations and models.
- Confusing area with perimeter.
- Treating the equals sign as an instruction to calculate.
- Assuming a correct answer proves understanding.